The performance of long-haul fibre-optic systems is essentially limited by the interplay of chromatic dispersion, fibre nonlinearity and noise. The rediscovery of coherent detection has paved the way for implementing sophisticated signal processing techniques in the electrical domain. However, without the availability of a mathematical model describing the input-output relationship of a nonlinear fibre link, effective compensation of transmission impairments is very difficult to achieve. Unfortunately, no exact analytical solution of the nonlinear Schrödinger equation, NLSE, describing the propagation of an optical field complex envelope is known in the presence of both chromatic dispersion and fibre nonlinearity.
Several approximations of a solution of the NLSE have been considered, including inverse scattering, back-propagation and various perturbation methods, namely Logarithmic Perturbation, LP, Regular Perturbation, RP, combined Regular-Logarithmic Perturbation, RLP, and Volterra Series Transfer Function, VSTF. The inverse scattering method is able to provide an exact solution only if the attenuation is negligible, which is not a practical case. It is for this reason that one of the most widely studied compensation strategies is digital back-propagation, which is based on the split-step Fourier method, the most widely used method for numerically solving the NLSE. An equaliser for an optical transmission system based on digital back-propagation is disclosed in WO 2010/094339.
Although both LP and RP can arbitrarily approximate the exact solution of the NLSE by using appropriately high orders, in practice only first-order solutions are acceptable for implementing reasonably efficient signal processing strategies. Both first-order RP, which has been shown to coincide with the third-order VSTF, and LP methods involve the same triple integral, but the LP method is significantly more accurate and its first-order solution remains accurate at higher input power levels, where the first-order RP solution breaks down. Yet, the LP solution may undergo numerical problems when the intensity approaches zero. For a continuous-wave signal, this difficulty can be overcome by using the RLP, but its extension to modulated signals is not simple. The VSTF involves use of a triple integral and the resulting computational complexity is too high to make practical implementation feasible.